--- Paul Kirby <[log in to unmask]> wrote:

> brings me to another question.  Can somebody, Steve perhaps, point
> in the
> direction of a discusion of risk.  As I understand it, our personal
> assesment of risk is coloured by two parts,  the probability of the
> event
> happening and its degree of awfulness. That is, many judge nuclear
> power as
> 'risky', not because accidents are likely but because they are
> scary.  I
> wear a life jacket in a boat not because I  am likely to fall in
> but
> because if I do I will drown.  (The obverse of this is our UK
> lottery where
> the chance of winning is minute but the reward is life changingly
> large).
> Statistical probability alone does not define what is individually
> perceived as being appropriate action.  Should we write off as
> wrong, or
> illinformed people who do not accept probability as being the sole
> measure
> of appropriate action?

Right the probability alone does not define the risk.  However, the
probabilities (or more correctly the probability densities) are a
necessary component in statistical decision making.  Without it you
cannot make the decision (or look at the result of such
calculations)...at least using the machinery of statistical decision
theory.

Here are a couple of links that you might find helpful.

http://darwin.eeb.uconn.edu/eeb310/lecture-notes/decision.pdf

http://pluto.huji.ac.il/~msby/modelsb-files/modelsb01.pdf

In the second one you'll see that one of the primary elements is the
loss function

L(theta,a)

theta is a random variable, and a is the action that is to be taken.

The action is determined by your decision rule d, which is a function
of x.  That is d: X -> A.  X is the result of when the random
variable takes on a specific value.

The risk function is defined as follows

R(theta,d) = E|theta [L(theta,d(x))]

Where the expectation is taken with regards to theta.

So if your loss function is such that you put a very large loss
associated with the negative outcome then even with a small
probability you might take "drastic" action.  So yeah, Paul, I'd
agree with your comment that there are "two parts" to the decision,
the probability (or density function) and the degree of loss.

Note that much of what I mentioned to Ray earlier about the problems
with aggregating prefrences (i.e. utility functions) holds here.  My
loss function might be quite different from another persons.  How to
aggregate up all these different loss functions (assuming you even
know what they are--an heroic assumption in itself) is quite
problematic.  I'd say that Arrow's Impossibility theorem would likely
rear its ugly head since a loss function can be "turned into" a
utility function simply by utilitizing the following operator

-L(.,.).

Steve
P.S.  The first link is much more chatty while the second is much
more mathematically rigorous.  Another aspect that can be brought
into this is the use of Bayesian methods where you update your prior
probability assessments as new information comes in.

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